Stability of periodic orbits in the elliptic, restricted three-body problem.

Abstract

A systematic study has been made of periodic orbits in the two-dimensional, elliptic, restricted three-body problem. All ranges of eccentricities, from 0 to 1, and mass-ratios, from 0 to J, have been investigated. Eleven hundred periodic orbits have been obtained. It is concluded that the elliptic problem behaves in a way which is completely different from the circular problem. The main difference is in the stability properties of the periodic orbits. Because of the nonexistence of the Jacobi integral (the elliptic problem is not conservative), the characteristic equation of the monodromy matrix does not have a pair of unit roots, in general. The stability is denned by two real numbers (stability indices) rather than one. Because of that, there are seven general classes of periodic orbits, according to their stability properties. The stability of the periodic orbits has been determined by numerically integrating the variational equations with a recurrent power series method. The results are in contrast with the circular problem, where there are only three classes of orbits (stability, even instability, and odd instability): in the elliptic problem there are one stable class and six unstable classes. The elliptic, restricted three-body problem can be considered as the prototype of all nonintegrable, nonconservative Hamiltonian systems, and in this paper, probably for the first time, a classification of the multipliers is given for these systems. I. Introduction